In this lesson, we'll be looking at Zeno of Elea, the ancient Greek philosopher who still manages to stump us in modern times. Learn about the man, his paradoxes, and test your knowledge with a quiz!

Biography

Is motion possible? Does a fast runner outrun a slow runner? The answers to these questions may seem obvious, but Zeno of Elea, an ancient Greek philosopher, presents us with a series of paradoxes that makes us question all of this.

Zeno of Elea was a Greek philosopher from the 5th century BCE who posed a series of paradoxes that continue to stump thinkers to this day. We don't know much about Zeno, so we have to rely on the accounts of Plato, Aristotle, and a couple other ancient writers. We do know Zeno visited Athens where he shared his book of paradoxes (which has not survived) with others, including the ancient Greek philosopher Socrates.

Zeno of Elea shows Youths the Doors to Truth and False, 16th century Spanish fresco

Zeno was a member of Paremenides's Eleatic school of Pre-Socratic philosophy. The Eleatics thought motion was impossible, believed that there are not multiple things but only one thing that exists, and doubted sense experience could reveal truth. It's thought that Zeno's paradoxes are meant to defend these Eleatic principles. These paradoxes were perhaps the first examples of the reduction ad absurdum technique, which deconstructed commonly held beliefs until all that was left was an absurd conclusion. The most famous and perplexing of his paradoxes are examined in more detail below.

The Paradox of Place

Zeno's first Paradox of Place makes us question how many things compose the universe. Zeno begins by stating that everything that exists--a dog, a chair, even a grain of sand--takes up space. If it takes up space, it must be in a place. Well that implies that place is also a thing that takes up space, which must also be in its own place that takes up space… and on and on we go. Therefore, according to Zeno, there must be an infinite number of things, spaces, and places. This concept of infinity was difficult for the ancient Greek world to comprehend, so the idea that there are infinite things in the universe was considered an absurd paradox.

The Achilles Paradox

Zeno's Achilles Paradox attempts to show that motion is impossible. Let's say a tortoise and the Greek hero Achilles were racing. Based on our experiences of how a man and a tortoise move, we would naturally assume that Achilles would win. But consider this: the tortoise has a head start and is constantly moving toward the finish line (Time 1, pictured). Before Achilles can pass the tortoise, Achilles has to reach the spot where the tortoise started, right (A2, Time 2, pictured)? But by the time Achilles reaches the tortoise's starting point, the tortoise has moved on to another location further down the track (T2, Time 2, pictured). So now Achilles has to reach the spot where the tortoise currently is. But by the time Achilles does that, the tortoise has yet again moved on (Time 3, pictured). So, the paradox is that Achilles will always be behind the tortoise and will lose the race.

Illustration of Achilles Paradox

The Dichotomy Paradox

The Dichotomy Paradox also shows that motion is an illusion. Let's say you were running a race. You see the finish line ahead. In order to get to the finish line, you know you first have to get to the halfway point. But before you can get to the halfway point, you have to get halfway to that halfway point. But before you get there, you have to get halfway to the halfway of the halfway point. And on it goes for eternity. Essentially, Zeno asserts that there are an infinite number of halfway points between any two locations. Before you can ever reach one location, you have to reach the halfway point first. But if there are an infinite number of halfway points, how can we possibly traverse an infinite number of locations?! 'That's absurd!' says Zeno. Therefore, motion is impossible, and you will never reach the finish line.

The Arrow Paradox

The goal of the Arrow Paradox is to show that what we think is motion is actually rest. Let's say you shoot an arrow at a target. The arrow then moves through the air, right? Not really, says Zeno. Zeno posited that time was a series of individual moments. In any given moment, the arrow can only be in one specific spot (because an object cannot be in two spots at once). But if the arrow is in one spot at any particular moment, that means the arrow is at rest at that moment, according to Zeno. So, if the arrow can only be in one spot at one time, how could it ever move? Therefore, the arrow must be at rest the whole time.

The Stadium Paradox

The Stadium Paradox says that an object can move half the distance and twice the distance at the same time, again making the notion of motion absurd. Picture 3 sets of blocks of the same size and the same distance apart, as shown. The A blocks are at rest. The B blocks are moving to the right, and the C blocks are moving to the left at the same speed. At Time 1, B3 is in the same column as A2 and C1. At Time 2, B3 is now in the same column as A3 and C3. So, B3 seemingly moved 2 spaces in relation to the C blocks but only 1 space in relation to the A blocks. Therefore, as Zeno would have us believe, the B blocks moved both 2 spaces and 1 space in the same amount of time, which is absurd.

Illustration of the Stadium Paradox

The Grain of Millet Paradox

You'd think that we could respond to Zeno that our experience solves these paradoxes. We know through experience that a fast runner passes a slow runner. Our senses tell us that an arrow really does move through the air to hit its target. You think you could tell Zeno this and he might concede, but you'd be wrong. Zeno argues that we can't trust our experience or our senses, and one final paradox, The Grain of Millet Paradox, shows us why.

While explaining the Grain of Millet Paradox, Zeno asks us to imagine dropping a big bag of millet grain. We would hear the thud as the bag hits the floor. But where exactly does the sound come from? It must come from the sound of every individual grain of millet making its own sound as it hits the floor. But if we drop only one grain of millet, we don't hear it, and we might (falsely) assume that an individual grain does not in fact make a sound. From Zeno's argument, though, we know that each individual grain of millet must make a sound, even if it is an infinitesimally small sound. Therefore, according to this paradox, we cannot trust our sense of hearing--or any of our senses for that matter.

Lesson Summary

So just when we think we might be able to use our senses to solve these paradoxes, Zeno's Grain of Millet paradox makes us doubt our own experiences. Through the Paradox of Place, Zeno makes us question just how many things and places compose the universe. Through the Achilles, Dichotomy, Arrow, and Stadium paradoxes, Zeno argues that motion is only an illusion.

Later philosophers, like Plato and Aristotle, offered their own solutions to the paradoxes. In addition, later mathematicians were able to solve the paradoxes through calculus. So even though there are solutions to Zeno's paradoxes, Zeno is credited with forcing mathematicians, scientists, philosophers, and anyone trying to get from point A to point B to look at the physical world in a different way.

Summary:

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